User:Ejc.cryptography

$$ a_{r} = \frac{\sqrt{2\pi}x_{w}}{L}[exp(-\frac{x_{w}^{2}}{2}(k_{w}+k_{r})^{2})+exp(-\frac{x_{w}^{2}}{2}(k_{w}-k_{r})^{2})]

$$

$$

\frac{GMm}{r^2} = \frac{mv^2}{r} \Rightarrow v = \sqrt{\frac{GM}{r}}

$$

$$

z_n = -\frac{x\times x_n+y\times y_n}{z}

$$

$$

\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}\cdot\begin{bmatrix} x_n\\ y_n\\ z_n\\ \end{bmatrix}=0

$$

$$

\begin{bmatrix} v_x\\ v_y\\ v_z\\ \end{bmatrix} = \frac{v}{\sqrt{x_n^2+y_n^2+z_n^2}}\begin{bmatrix} x_n\\ y_n\\ z_n\\ \end{bmatrix}

$$

$$

x\times x_n + y\times y_n + z\times z_n = 0

$$

$$

y_1={g}^{x_1} \pmod{p}

$$

$$

y_2={g}^{x_2} \pmod{p}

$$

$$

s={y_1}^{x_2} \pmod{p}

$$

$$

s={y_2}^{x_1} \pmod{p}

$$

$$

s=(g^{x_1})^{x_2} \pmod{p}

$$

$$

s=(g^{x_2})^{x_1} \pmod{p}

$$

$$

s=g^{x_1 x_2} \pmod{p}

$$

$$ p = 53 \, $$

$$ g = 18 \, $$

$$ x_1 = 8 \, $$

$$ y_1=18^8 \pmod{53}=24 \, $$

$$ x_2=11 \, $$

$$ y_2=18^{11} \pmod{53}=48 \, $$

$$ s=24^{11} \pmod{53}=15 \, $$

$$ s=48^{8} \pmod{53}=15 \, $$

$$ 24=18^{x_1} \pmod{53} \, $$